Page 151 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

142 Angular momentum
2ð ð

Y (è, j)Y lm (è, ö) sin è dè dj
lm
0 0
ð 2ð

ˆ È (è)È lm (è) sin è dè Ö (j)Ö m (j)dj ˆ 1
lm
m
0 0
where the è- and j-dependent parts of the volume element dô are included in
the integration. For convenience, we require that each of the two factors È lm (è)
and Ö m (j) be normalized. Writing Ö m (j)as
Ö m (j) ˆ Ae imj
we ®nd that
2ð 2ð

(Ae imj imj )dj ˆjAj 2 dj ˆ 1
) (Ae
0 p 0

iá
A ˆ e = 2ð
giving
1 imj
Ö m (j) ˆ p  e (5:40)
2ð
iá
where we have arbitrarily set á equal to zero in the phase factor e associated
with the normalization constant.
The function È l,ÿl (è) is given by equation (5.39) and the value of the
constant of integration A l is determined by the normalization condition
ð ð

2 2l‡1
[È l,ÿl (è)] È l,ÿl (è) sin è dè ˆjA l j sin è dè ˆ 1 (5:41)
0 0
We need to evaluate the integral I l
ð ÿ1 1

2 l
2 l
I l sin 2l‡1 è dè ˆÿ (1 ÿ ì ) dì ˆ (1 ÿ ì ) dì
0 1 ÿ1
where we have de®ned the variable ì by the relation
ì cos è (5:42)
so that
2
2
1 ÿ ì ˆ sin è, dì ˆÿsin è dè
The integral I l may be transformed as follows

1 1 1 ì
2 l
2 lÿ1 2
2 lÿ1
I l ˆ (1 ÿ ì ) dì ÿ (1 ÿ ì ) ì dì ˆ I lÿ1 ‡ d(1 ÿ ì )
ÿ1 ÿ1 ÿ1 2l
1 2 l

(1 ÿ ì ) 1
ˆ I lÿ1 ÿ dì ˆ I lÿ1 ÿ I l
ÿ1 2l 2l
where we have integrated by parts and noted that the integrated term vanishes.
Solving for I l , we obtain a recurrence relation for the integral